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Half-life and doubling time

The half-life is the time it takes for a decreasing quantity to be halved, and the doubling time is the corresponding time for a growing quantity. Both follow directly from the growth rate $k$ .

T = \frac{\ln 2}{|k|}

half-life or doubling time

Symbols

$T$	half-life or doubling time	time
$k$	growth rate	1/time

Example

A radioactive source decays by 12% per year: $k=\ln 0.88 \approx -0.128$ , so $T=\dfrac{\ln 2}{0.128}\approx 5.42$ years.

The formula holds no matter where you start on the curve - the half-life is always the same for exponential decay.

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Part of Foundations of Mathematics: Exponential and logarithmic functions.